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Finding freq., Wavelentgh, Phase Velocity, and attenuation constant

  1. Jan 15, 2008 #1
    1. The problem statement, all variables and given/known data

    Given...

    [tex]v\left(z,\,t\right)\,=\,5\,e^{-\alpha\,z}\,sin\left(4\pi\,\times\,10^9\,t\,-\,20\pi\,z\right)[/tex]

    where z is distance (m), find...

    (a) Frequency

    (b) Wavelength

    (c) Phase Velocity

    (d) At z = 2m, the amplitude is 1 [V], Find the attenuation constant ([itex]\alpha[/itex]).



    2. Relevant equations

    [tex]f\,=\,\frac{1}{T}[/tex]

    [tex]y\left(x,\,t\right)\,=\,A\,cos\left(\frac{2\pi\,t}{T}\,-\,\frac{2\pi\,x}{\lambda}\,+\,\phi_0\right)[/tex]

    [tex]u_p\,=\,f\,\lambda[/tex]



    3. The attempt at a solution


    (a)

    Using the first term ([itex]\frac{2\pi\,t}{T}[/itex]) in the argument to the cosine in the general form above...

    [tex]\frac{2\pi}{T}\,=\,4\pi\,\times\,10^9\,\,\longrightarrow\,\,T\,=\,\frac{2\pi}{4\pi\,\times\,10^9}\,=\,0.5\,\times\,10^{-9}[/tex]

    [tex]f\,=\,\frac{1}{T}\,=\,\frac{1}{0.5\,\times\,10^{-9}}\,=\,2\,\times\,10^9\,=\,2\,Ghz[/tex]


    (b)

    Using the second term ([itex]-\,\frac{2\pi\,x}{\lambda}[/itex]) in the argument to the cosine in the general form above...

    [tex]\frac{2\pi}{\lambda}\,=\,20\pi\,\,\longrightarrow\,\,\lambda\,=\,\frac{2\pi}{20\pi}\,=\,\frac{1}{10}\,=\,0.1\,m[/tex]


    (c)

    [tex]u_p\,=\,f\,\lambda\,=\,\left(2\,\times\,10^9\right)\,(0.1)\,=\,200,000,000\,\frac{m}{s}[/tex]


    (d)

    [tex]1\,=\,5\,e^{-2\,\alpha}\,sin\left(4\pi\,\times\,10^9\,t\,-\,40\pi\right)[/tex]

    [tex]5\,e^{-2\alpha}\,=\,1\,\,\longrightarrow\,\,-2\alpha\,=\,ln\left(\frac{1}{5}\right)\,\,\longrightarrow\,\,\alpha\,=\,0.8047[/tex]

    Right?
     
    Last edited: Jan 15, 2008
  2. jcsd
  3. Jan 16, 2008 #2
    How did you eliminate 't' in part d?
     
  4. Jan 16, 2008 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    There is no "t" in (d). The "attenuation" constant is the rate at which the magnitude of the wave degrades- and that depends entirely upon the coefficient of the cosine term, [itex]5e^{-\alpha z}[/itex]. And here, we are given that z= 2.
     
  5. Jan 16, 2008 #4
    What about the sin term. 1= 5xexp(-alpha x z) x sin term which contains t?
     
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